im quite baffled by this question i was recently asked
so why does e^ (i*pi) = -1?

Any information is greatly appreciated!

Let's suppose that Leonhard Euler has been a poet and not a mathematician, so that we ignore his formula. In that case the answer to Your question may derive from the solution of the differential equation...

$\displaystyle z^{'}= i\ z\ , \ z(0)=1$ (1)

The (1) is the trajectory of a point the velocity of which is its postion rotated by $\displaystyle \displaystyle \frac{pi}{2}$ so that we have an 'uniform circular move'. The solution of (1) is $\displaystyle z=e^{i\ t}$ so that the question is: where will be the point at the time $\displaystyle t= \pi$?...

Let's suppose that Leonhard Euler has been a poet and not a mathematician, so that we ignore his formula. In that case the answer to Your question may derive from the solution of the differential equation...

$\displaystyle z^{'}= i\ z\ , \ z(0)=1$ (1)

The (1) is the trajectory of a point the velocity of which is its postion rotated by $\displaystyle \displaystyle \frac{pi}{2}$ so that we have an 'uniform circular move'. The solution of (1) is $\displaystyle z=e^{i\ t}$ so that the question is: where will be the point at the time $\displaystyle t= \pi$?...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

In fact, any complex number can be written as $\displaystyle \displaystyle z = r\cos{x} + i\,r\sin{x}$

so it is quite easy to see that $\displaystyle \displaystyle \frac{dz}{dx} = -r\sin{x} + i\,r\cos{x}$